Modular Representation Theory and Categorification with Applications

模块化表示理论及其分类及其应用

• 批准号: 2101791
• 负责人: Alexander Kleshchev
• 金额: $ 27.05万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101791&HistoricalAwards=false
• 关键词: Modular; Representation; Theory; Categorification; Applications

Groups are mathematical objects arising in the study of symmetry. The main foci of this project are some of the most fundamental and universal examples of groups: symmetric groups, which arise as symmetries of finite sets, and general linear groups, which arise as symmetries of finite-dimensional vector spaces. Representation theory studies groups via their actions on other mathematical objects, such as vector spaces. Very informally, representations of a group are snap-shots of the group taken from different directions. In the past few years, the idea of categorification has become very important and has led to the development of higher representation theory. This involves actions of groups on higher mathematical structures called categories, utilizing not only the relations between these structures (functors) but also relations between the relations (natural transformations). In particular, Khovanov-Lauda-Rouquier (KLR) algebras encode higher symmetries underlying a large part of representation theory, including classical objects like symmetric and general linear groups. The goal of this project is to further build the theory of these and other algebras and apply it to improve our understanding of the classical objects of group theory. The research in this project has potential future broader impacts in computer science and theoretical physics. More directly this award will have important educational impact through the training of graduate students and mentoring young researchers in this area. In more detail, this project is concerned with several diverse projects in representation theory of Lie algebras, finite groups, and related objects such as Hecke algebras, quantum groups, Schur algebras and KLR algebras. Our perspective draws on recent advances in higher representation theory, namely categorification, with various diagrammatically defined monoidal categories and 2-categories playing a prominent role. On the other hand, many applications are to classical problems in representation theory such as block theory of finite groups and Schur algebras as well as structure theory of finite groups. We will study local description of blocks of Schur algebras up to derived equivalence, cuspidal algebras for KLR algebras, thick Heisenberg categorification, super Kac-Moody 2-categories and applications to blocks of double covers of symmetric groups, homological properties of KLR algebras, decomposition numbers, and irreducible restrictions from quasi-simple groups to subgroups. The project will have applications to other areas of mathematics including finite group theory (and its applications), Lie theory, combinatorics, representation theory, knot theory and category theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

组是对称性研究中产生的数学对象。该项目的主要焦点是群体中一些最基本和普遍的示例：对称群体（作为有限集的对称性）和一般线性群体的对称群体，这些群体是有限维矢量空间的对称性。表示理论通过对其他数学对象（例如向量空间）的行为进行研究。非常非正式的是，一个小组的表示是从不同方向拍摄的小组的快照。在过去的几年中，分类的想法变得非常重要，并导致了更高代表理论的发展。这涉及组对较高数学结构的行动称为类别，不仅利用这些结构（函数）之间的关系，还利用关系之间的关系（自然变换）。特别是，Khovanov-Lauda-Rouquier（KLR）代数编码了代表理论的大部分代表理论的较高对称性，包括对称对象和一般线性群。该项目的目的是进一步构建这些和其他代数的理论，并将其应用于我们对群体理论的经典对象的理解。该项目中的研究对计算机科学和理论物理学有潜在的更广泛的影响。更直接地，该奖项将通过培训研究生并指导该领域的年轻研究人员而产生重要的教育影响。 更详细地，该项目涉及谎言代数，有限群体和相关对象的代表理论中的几个不同项目，例如Hecke代数，量子群，Schur代数和KLR代数。我们的观点借鉴了高度代表理论的最新进展，即分类，具有各种示意性的单体类别和两类发挥着重要作用。另一方面，许多应用是代表理论中的经典问题，例如有限群体和Schur代数的块理论以及有限群体的结构理论。 We will study local description of blocks of Schur algebras up to derived equivalence, cuspidal algebras for KLR algebras, thick Heisenberg categorification, super Kac-Moody 2-categories and applications to blocks of double covers of symmetric groups, homological properties of KLR algebras, decomposition numbers, and irreducible restrictions from quasi-simple groups to subgroups.该项目将在数学的其他领域中应用，包括有限的群体理论（及其应用），谎言理论，组合理论，代表理论，结理论和类别理论。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的审查标准来通过评估来支持的。